Curl bounds grad on SO(3)
ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 1, pp. 148-159.

Let F p GL (3) be the plastic deformation from the multiplicative decomposition in elasto-plasticity. We show that the geometric dislocation density tensor of Gurtin in the form Curl [F p ]·(F p ) T applied to rotations controls the gradient in the sense that pointwise RC 1 ( 3 , SO (3)): Curl [R]·R T 𝕄 3×3 2 1 2DR 27 2 . This result complements rigidity results [Friesecke, James and Müller, Comme Pure Appl. Math. 55 (2002) 1461-1506; John, Comme Pure Appl. Math. 14 (1961) 391-413; Reshetnyak, Siberian Math. J. 8 (1967) 631-653)] as well as an associated linearized theorem saying that AC 1 ( 3 ,𝔰𝔬(3)): Curl [A] 𝕄 3×3 2 1 2DA 27 2 = axl [A] 9 2 .

DOI : 10.1051/cocv:2007050
Classification : 74A35, 74E15, 74G65, 74N15, 53AXX, 53B05
Mots clés : rotations, polar-materials, microstructure, dislocation density, rigidity, differential geometry, structured continua
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Münch, Ingo; Neff, Patrizio. Curl bounds grad on SO(3). ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 1, pp. 148-159. doi : 10.1051/cocv:2007050. http://www.numdam.org/articles/10.1051/cocv:2007050/

[1] S. Aubry and M. Ortiz, The mechanics of deformation-induced subgrain-dislocation structures in metallic crystals at large strains. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 459 (2003) 3131-3158. | MR | Zbl

[2] B.A. Bilby, R. Bullough and E. Smith, Continuous distributions of dislocations: a new application of the methods of non-riemannian geometry. Proc. Roy. Soc. London, Ser. A 231 (1955) 263-273. | MR

[3] E. Cartan, Leçons sur la géometrie des espaces de Riemann. Gauthier-Villars, Paris (1928). | JFM | MR | Zbl

[4] P. Cermelli and M.E. Gurtin, On the characterization of geometrically necessary dislocations in finite plasticity. J. Mech. Phys. Solids 49 (2001) 1539-1568. | Zbl

[5] S. Conti and M. Ortiz, Dislocation microstructures and the effective behavior of single crystals. Arch. Rat. Mech. Anal. 176 (2005) 103-147. | MR | Zbl

[6] A. Einstein, Relativity: The Special and General Theory. Crown, New-York (1961). | JFM | Zbl

[7] J.D. Eshelby, The continuum theory of lattice defects, volume III of Solid state Physics. Academic Press, New-York (1956).

[8] G. Friesecke, R.D. James and S. Müller, A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. 55 (2002) 1461-1506. | MR | Zbl

[9] M.E. Gurtin, An Introduction to Continuum Mechanics, Mathematics in Science and Engineering 158. Academic Press, London, 1st edn. (1981). | MR | Zbl

[10] M.E. Gurtin, On the plasticity of single crystals: free energy, microforces, plastic-strain gradients. J. Mech. Phys. Solids 48 (2000) 989-1036. | MR | Zbl

[11] J.P. Hirth and J. Lothe, Theory of Dislocations. McGraw-Hill, New-York (1968).

[12] F. John, Rotation and strain. Comm. Pure Appl. Math. 14 (1961) 391-413. | MR | Zbl

[13] J. Jost, Riemannian Geometry. Springer-Verlag (2002). | MR

[14] K. Kondo, Geometry of elastic deformation and incompatibility, in Memoirs of the Unifying Study of the Basic Problems in Engineering Science by Means of Geometry, volume 1, Division C, K. Kondo Ed., Gakujutsu Bunken Fukyo-Kai (1955) 361-373. | MR | Zbl

[15] E. Kröner, Der fundamentale Zusammenhang zwischen Versetzungsdichte und Spannungsfunktion. Z. Phys. 142 (1955) 463-475. | MR | Zbl

[16] E. Kröner, Kontinuumstheorie der Versetzungen und Eigenspannungen, Ergebnisse der Angewandten Mathematik 5. Springer, Berlin (1958). | MR | Zbl

[17] E. Kröner, Allgemeine Kontinuumstheorie der Versetzungen und Eigenspannungen. Arch. Rat. Mech. Anal. 4 (1960) 273-334. | MR | Zbl

[18] E. Kröner and A. Seeger, Nichtlineare Elastizitätstheorie der Versetzungen und Eigenspannungen. Arch. Rat. Mech. Anal. 3 (1959) 97-119. | MR | Zbl

[19] D. Kuhlmann-Wilsdorf, Theory of plastic deformation: properties of low energy dislocation structures. Mat. Sci. Eng. A113 (1989) 1.

[20] E.H. Lee, Elastic-plastic deformation at finite strain. J. Appl. Mech. 36 (1969) 1-6. | Zbl

[21] A. Mielke and S. Müller, Lower semi-continuity and existence of minimizers in incremental finite-strain elastoplasticity. ZAMM 86 (2006) 233-250. | MR | Zbl

[22] T. Mura, Micromechanics of defects in solids. Kluwer Academic Publishers, Boston (1987). | Zbl

[23] F.R.N. Nabarro, Theory of crystal dislocations. Oxford University Press, Oxford (1967).

[24] J. Necas and I. Hlavacek, Mathematical theory of elastic and elastico-plastic bodies: An introduction. Elsevier, Amsterdam (1981). | MR | Zbl

[25] P. Neff, On Korn's first inequality with nonconstant coefficients. Proc. Roy. Soc. Edinb. A 132 (2002) 221-243. | MR | Zbl

[26] J.F. Nye, Some geometrical relations in dislocated crystals. Acta Metall. 1 (1953) 153-162.

[27] M. Ortiz and E.A. Repetto, Nonconvex energy minimization and dislocation structures in ductile single crystals. J. Mech. Phys. Solids 47 (1999) 397-462. | MR | Zbl

[28] M. Ortiz, E.A. Repetto and L. Stainier, A theory of subgrain dislocation structures. J. Mech. Phys. Solids 48 (2000) 2077-2114. | MR | Zbl

[29] G.P. Parry and M. Silhavy, Elastic scalar invariants in the theory of defective crystals. R. Soc. Lond. Proc., Ser. A, Math. Phys. Eng. Sci. 455 (1999) 4333-4346. | MR | Zbl

[30] Yu.G. Reshetnyak, Liouville's theorem on conformal mappings for minimal regularity assumptions. Siberian Math. J. 8 (1967) 631-653. | MR | Zbl

[31] B. Svendsen, Continuum thermodynamic models for crystal plasticity including the effects of geometrically necessary dislocations. J. Mech. Phys. Solids 50 (2002) 1297-1329. | MR | Zbl

[32] R.M. Wald, General Relativity. University of Chicago Press, Chicago (1984). | MR | Zbl

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